概率论：分赌注问题理论分析+matlab实现 177.81KB

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问题描述：水平相同的两个赌徒A和B，约定先胜t局的人赢得赌注，在赌注中的某时刻，两赌徒中止赌博，此时A胜r局，B胜s局，应如何分配赌注？分析解决：利用概率论相关知识，将具体问题抽象为数学问题，计算出理论结果。再利用matlab进行题目仿真，经多次仿真得到仿真数据。其中，给出了具体推导过程，matlab源代码以及流程图。重要性：赌注问题称为概率论的起源。当荷兰数学家惠更斯（Huygens,C.）到巴黎时，听说费马和帕斯卡在研究赌注问题，也进行了研究，并在1657年撰写了《论赌博中的计算》一书，提出数学期望的概念，推动了概率论的发展。

<link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/base.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css" rel="stylesheet"/><link href="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89601398/raw.css" rel="stylesheet"/><div id="sidebar" style="display: none"><div id="outline"></div></div><div class="pf w0 h0" data-page-no="1" id="pf1"><div class="pc pc1 w0 h0"><img alt="" class="bi x0 y0 w1 h1" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89601398/bg1.jpg"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">赌注问题</div><div class="t m0 x2 h3 y2 ff2 fs1 fc0 sc1 ls0 ws0">一问题描<span class="_ _0"></span>述及背景</div><div class="t m0 x3 h4 y3 ff1 fs2 fc0 sc0 ls0 ws0">水平相同的两个赌徒<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>和<span class="_ _1"> </span><span class="ff3">B</span>，<span class="_ _2"></span>约定先胜<span class="_ _1"> </span><span class="ff3">t<span class="_ _1"> </span></span>局的人赢得赌注，<span class="_ _2"></span>在赌</div><div class="t m0 x2 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">注中的某时刻，两赌徒<span class="_ _0"></span>中止赌博，此时<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>胜<span class="_ _1"> </span><span class="ff3">r<span class="_ _1"> </span></span>局<span class="_ _0"></span>，<span class="ff3">B<span class="_ _1"> </span></span>胜<span class="_ _1"> </span><span class="ff3">s<span class="_ _1"> </span></span>局，应如何</div><div class="t m0 x2 h4 y5 ff1 fs2 fc0 sc0 ls0 ws0">分配赌注？</div><div class="t m0 x4 h5 y6 ff3 fs3 fc0 sc0 ls0 ws0">-------<span class="_ _3"> </span><span class="ff1">这<span class="_ _3"> </span>个<span class="_ _3"> </span>问<span class="_ _3"> </span>题<span class="_ _3"> </span>通<span class="_ _3"> </span>常<span class="_ _3"> </span>称<span class="_ _3"> </span>为<span class="_ _3"> </span>点<span class="_ _3"> </span>数<span class="_ _3"> </span>问<span class="_ _3"> </span>题<span class="_ _3"> </span>，<span class="_ _3"> </span>由<span class="_ _3"> </span>嗜<span class="_ _4"> </span>好<span class="_ _3"> </span>赌<span class="_ _4"> </span>博<span class="_ _3"> </span>的<span class="_ _3"> </span>法<span class="_ _3"> </span>国<span class="_ _3"> </span>学<span class="_ _3"> </span>者<span class="_ _3"> </span>梅<span class="_ _3"> </span>雷</span></div><div class="t m0 x2 h5 y7 ff3 fs3 fc0 sc0 ls0 ws0">(Meray,H.C.R)<span class="ff1">于<span class="_ _5"> </span></span>1654<span class="_ _5"> </span><span class="ff1">年向法国<span class="_ _0"></span>数学<span class="_ _0"></span>家帕斯<span class="_ _0"></span>卡</span>(Pascal,B.)<span class="_ _0"></span><span class="ff1">提出<span class="_ _0"></span>的。为<span class="_ _0"></span>此帕<span class="_ _0"></span>斯卡<span class="_ _0"></span>和法</span></div><div class="t m0 x2 h5 y8 ff1 fs3 fc0 sc0 ls0 ws0">国数<span class="_ _0"></span>学家<span class="_ _0"></span>费马<span class="_ _0"></span>（<span class="ff3">Fer-mat,P.de<span class="_ _0"></span></span>）于<span class="_ _5"> </span><span class="ff3">1654<span class="_ _5"> </span></span>年<span class="_ _5"> </span><span class="ff3">7<span class="_ _5"> </span></span>月到<span class="_ _5"> </span><span class="ff3">10<span class="_"> </span></span>月<span class="_ _0"></span>之间<span class="_ _0"></span>进<span class="_ _0"></span>行了一<span class="_ _0"></span>系<span class="_ _0"></span>列通<span class="_ _0"></span>信讨</div><div class="t m0 x2 h5 y9 ff1 fs3 fc0 sc0 ls0 ws0">论。赌注问题称为<span class="_ _0"></span>概率论的起源。当<span class="_ _0"></span>荷兰数学家惠更斯<span class="_ _0"></span>（<span class="ff3">Huygens,C.</span>）到巴黎时<span class="_ _0"></span>，</div><div class="t m0 x2 h5 ya ff1 fs3 fc0 sc0 ls0 ws0">听说费马和帕斯卡在<span class="_ _0"></span>研究赌注问题，也进<span class="_ _0"></span>行了研究，并在<span class="_ _5"> </span><span class="ff3">1657<span class="_ _5"> </span></span>年撰写了《论赌</div><div class="t m0 x2 h5 yb ff1 fs3 fc0 sc0 ls0 ws0">博中的计算》一书，提出数学期望的概念，推动了概率论的发展。</div><div class="t m0 x2 h3 yc ff2 fs1 fc0 sc1 ls0 ws0">二问题分<span class="_ _0"></span>析与解决</div><div class="t m0 x3 h4 yd ff1 fs2 fc0 sc0 ls0 ws0">两赌徒水平相同，<span class="_ _6"></span>所以每人每局获胜的概率均为<span class="ff3"> </span></div><div class="c x5 ye w2 h6"><div class="t m1 x6 h7 yf ff4 fs4 fc0 sc0 ls0 ws0">1</div></div><div class="c x5 ye w2 h8"><div class="t m1 x6 h7 y10 ff4 fs4 fc0 sc0 ls0 ws0">2</div></div><div class="t m0 x7 h4 yd ff3 fs2 fc0 sc0 ls0 ws0"> <span class="_ _7"> </span><span class="ff1">，<span class="_ _6"></span><span class="ff3">A<span class="_ _1"> </span><span class="ff1">获胜与<span class="_ _1"> </span></span>B</span></span></div><div class="t m0 x2 h4 y11 ff1 fs2 fc0 sc0 ls0 ws0">获胜为对立事件，所以求最后赌注的分配就是求<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>与<span class="_ _1"> </span><span class="ff3">B<span class="_ _1"> </span></span>获胜的概率。</div><div class="t m0 x3 h4 y12 ff1 fs2 fc0 sc0 ls0 ws0">假设先前获胜局数<span class="_ _1"> </span><span class="ff3">r>s</span>，</div><div class="t m0 x3 h4 y13 ff1 fs2 fc0 sc0 ls0 ws0">若<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>最后获胜，那么最少需要再进行<span class="_ _1"> </span><span class="ff3">t-r<span class="_ _1"> </span></span>局（之后<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>一直连胜，</div><div class="t m0 x2 h4 y14 ff1 fs2 fc0 sc0 ls0 ws0">直至最终获胜）<span class="_ _8"></span>，最多需要进行<span class="_ _1"> </span><span class="ff3">2t-r-s-1<span class="_ _1"> </span></span>局（<span class="ff3">A<span class="_ _1"> </span></span>与<span class="_ _1"> </span><span class="ff3">B<span class="_ _1"> </span></span>各差一局获胜）。</div><div class="t m0 x3 h4 y15 ff1 fs2 fc0 sc0 ls0 ws0">假设<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>在第<span class="_ _1"> </span><span class="ff3">n<span class="_ _1"> </span></span>局时最终获胜。</div><div class="t m0 x3 h4 y16 ff1 fs2 fc0 sc0 ls0 ws0">则<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>在前<span class="_ _0"></span><span class="ff3">(n-1)</span>局获<span class="_ _0"></span>胜了<span class="ff3">(t-r-1)</span>局<span class="_ _0"></span>，且在<span class="_ _0"></span>第<span class="_ _1"> </span><span class="ff3">n<span class="_ _1"> </span></span>局<span class="_ _0"></span>获得胜利<span class="_ _0"></span>，则<span class="_ _1"> </span><span class="ff3">A<span class="_ _7"> </span></span>在</div><div class="t m0 x2 h4 y17 ff1 fs2 fc0 sc0 ls0 ws0">前<span class="ff3">(n-1)</span>局获胜局数<span class="_ _1"> </span><span class="ff3">X<span class="_ _1"> </span></span>满足二项分布</div><div class="t m0 x8 h4 y18 ff3 fs2 fc0 sc0 ls0 ws0">X~B<span class="ff1">（</span>n-1,</div><div class="c x9 ye w2 h9"><div class="t m1 x6 h7 y19 ff4 fs4 fc0 sc0 ls0 ws0">1</div></div><div class="c x9 ye w2 ha"><div class="t m1 x6 h7 y1a ff4 fs4 fc0 sc0 ls0 ws0">2</div></div><div class="t m0 xa h4 y18 ff1 fs2 fc0 sc0 ls0 ws0">）</div><div class="t m0 x2 h4 y1b ff1 fs2 fc0 sc0 ls0 ws0">所以获胜在前<span class="ff3">(n-1)</span>局获胜了<span class="ff3">(t-r-1)</span>的概率为</div></div><div class="pi" data-data='{"ctm":[1.611830,0.000000,0.000000,1.611830,0.000000,0.000000]}'></div></div><div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="/image.php?url=https://csdnimg.cn/release/download_crawler_static/89601398/bg2.jpg"><div class="c xb ye w3 hb"><div class="t m0 xc hc y1c ff4 fs2 fc0 sc0 ls0 ws0">P(X</div></div><div class="c xd ye w4 hb"><div class="t m0 xc hc y1c ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c xe ye w5 hb"><div class="t m0 xc hc y1c ff5 fs2 fc0 sc0 ls0 ws0">t−r−1)</div></div><div class="c xf ye w4 hb"><div class="t m0 xc hc y1c ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x10 ye w6 hd"><div class="t m0 xc hc y1d ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x11 y1e w7 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟−1</div></div><div class="c x11 y20 w8 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x12 ye w9 hb"><div class="t m0 xc hc y1c ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x13 ye wa hf"><div class="t m0 xc hc y21 ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x13 ye wa h10"><div class="t m0 xc hc y22 ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x14 ye w9 hb"><div class="t m0 xc hc y1c ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x15 y23 w8 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="t m0 x2 h4 y24 ff1 fs2 fc0 sc0 ls0 ws0">所以<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>在第<span class="_ _1"> </span><span class="ff3">n<span class="_ _1"> </span></span>局最终获胜的概率为</div><div class="c x16 ye wb h11"><div class="t m0 xc hc y25 ff4 fs2 fc0 sc0 ls0 ws0">P</div></div><div class="c x17 y26 wc he"><div class="t m1 x6 h7 y1f ff4 fs4 fc0 sc0 ls0 ws0">An</div></div><div class="c xb ye w4 h12"><div class="t m0 xc hc y27 ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x18 ye w6 h11"><div class="t m0 xc hc y25 ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x19 y28 w7 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟−1</div></div><div class="c x19 y29 w8 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x1a ye w9 h12"><div class="t m0 xc hc y27 ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x1b ye wa h13"><div class="t m0 xc hc y2a ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x1b ye wa h14"><div class="t m0 xc hc y2b ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x1c ye w9 h12"><div class="t m0 xc hc y27 ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x1d y2c w8 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x1e ye wd h12"><div class="t m0 xc hc y27 ff4 fs2 fc0 sc0 ls0 ws0">×</div></div><div class="c x1f ye wa h13"><div class="t m0 xc hc y2a ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x1f ye wa h14"><div class="t m0 xc hc y2b ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x20 ye w4 h12"><div class="t m0 xc hc y27 ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x21 ye w6 h11"><div class="t m0 xc hc y25 ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x22 y28 w7 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟−1</div></div><div class="c x22 y29 w8 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x23 ye wa h15"><div class="t m0 xc hc y2d ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x23 ye wa h16"><div class="t m0 xc hc y2e ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x24 y2f we he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="t m0 x2 h4 y30 ff1 fs2 fc0 sc0 ls0 ws0">而在第<span class="ff3">(t-r)</span>局到第<span class="ff3">(2t-r-s-1)</span>局之中，<span class="ff3">A<span class="_ _1"> </span></span>均有可能获胜，</div><div class="t m0 x2 h4 y31 ff1 fs2 fc0 sc0 ls0 ws0">故<span class="_ _1"> </span><span class="ff3">A<span class="_ _1"> </span></span>最终获胜的概率为</div><div class="c x18 ye wb h17"><div class="t m0 xc hc y32 ff4 fs2 fc0 sc0 ls0 ws0">P</div></div><div class="c x19 ye wf h18"><div class="t m1 x6 h7 y33 ff4 fs4 fc0 sc0 ls0 ws0">A</div></div><div class="c x25 ye w4 h19"><div class="t m0 xc hc y34 ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x26 y35 w10 he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">2𝑡−𝑟−𝑠−1</div></div><div class="c x27 ye we h1a"><div class="t m1 x6 h7 y36 ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="c x1b ye w11 h1a"><div class="t m1 x6 h7 y36 ff4 fs4 fc0 sc0 ls0 ws0">=</div></div><div class="c x28 ye w12 h1a"><div class="t m1 x6 h7 y36 ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟</div></div><div class="c x10 ye w6 h1b"><div class="t m0 xc hc y37 ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x11 ye w7 h1c"><div class="t m1 x6 h7 y38 ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟−1</div></div><div class="c x11 ye w8 h1d"><div class="t m1 x6 h7 y39 ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x12 ye w9 h19"><div class="t m0 xc hc y34 ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x13 ye wa h1e"><div class="t m0 xc hc y3a ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x13 ye wa h1f"><div class="t m0 xc hc y3b ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x14 ye w9 h19"><div class="t m0 xc hc y34 ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x15 y3c we he"><div class="t m1 x6 h7 y1f ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="t m0 x2 h4 y3d ff1 fs2 fc0 sc0 ls0 ws0">所以<span class="_ _1"> </span><span class="ff3">B<span class="_ _1"> </span></span>最终获胜的概率为</div><div class="c x29 ye wb h20"><div class="t m0 xc hc y3e ff4 fs2 fc0 sc0 ls0 ws0">P</div></div><div class="c x2a ye w13 h21"><div class="t m1 x6 h7 y3f ff4 fs4 fc0 sc0 ls0 ws0">B</div></div><div class="c x2b ye w4 h20"><div class="t m0 xc hc y3e ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x2c ye w14 h20"><div class="t m0 xc hc y3e ff5 fs2 fc0 sc0 ls0 ws0">1−P</div></div><div class="c xe ye wf h21"><div class="t m1 x6 h7 y3f ff4 fs4 fc0 sc0 ls0 ws0">A</div></div><div class="c x2d ye w4 h22"><div class="t m0 xc hc y40 ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x8 ye w15 h22"><div class="t m0 xc hc y40 ff5 fs2 fc0 sc0 ls0 ws0">1−</div></div><div class="c x2e ye w10 h23"><div class="t m1 x6 h7 y41 ff5 fs4 fc0 sc0 ls0 ws0">2𝑡−𝑟−𝑠−1</div></div><div class="c x2f ye we h24"><div class="t m1 x6 h7 y42 ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="c x30 ye w11 h24"><div class="t m1 x6 h7 y42 ff4 fs4 fc0 sc0 ls0 ws0">=</div></div><div class="c x31 ye w12 h24"><div class="t m1 x6 h7 y42 ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟</div></div><div class="c x32 ye w6 h25"><div class="t m0 xc hc y43 ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x33 ye w7 h26"><div class="t m1 x6 h7 y44 ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑟−1</div></div><div class="c x33 ye w8 h27"><div class="t m1 x6 h7 y45 ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x34 ye w9 h22"><div class="t m0 xc hc y40 ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x35 ye wa h28"><div class="t m0 xc hc y46 ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x35 ye wa h29"><div class="t m0 xc hc y47 ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x36 ye w9 h22"><div class="t m0 xc hc y40 ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x37 ye we h2a"><div class="t m1 x6 h7 y48 ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="t m0 x2 h4 y49 ff1 fs2 fc0 sc0 ls0 ws0">同理可得</div><div class="t m0 x2 h4 y4a ff1 fs2 fc0 sc0 ls0 ws0">当<span class="_ _1"> </span><span class="ff3">r<s<span class="_ _1"> </span></span>时</div><div class="c x38 ye wb h2b"><div class="t m0 xc hc y4b ff4 fs2 fc0 sc0 ls0 ws0">P</div></div><div class="c x39 ye wf h2c"><div class="t m1 x6 h7 y4c ff4 fs4 fc0 sc0 ls0 ws0">A</div></div><div class="c x3a ye w4 h2d"><div class="t m0 xc hc y4d ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x3b ye w15 h2d"><div class="t m0 xc hc y4d ff5 fs2 fc0 sc0 ls0 ws0">1−</div></div><div class="c x27 ye w10 h2e"><div class="t m1 x6 h7 y4e ff5 fs4 fc0 sc0 ls0 ws0">2𝑡−𝑟−𝑠−1</div></div><div class="c x3c ye we h2f"><div class="t m1 x6 h7 y4f ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="c x1c ye w11 h2f"><div class="t m1 x6 h7 y4f ff4 fs4 fc0 sc0 ls0 ws0">=</div></div><div class="c x3d ye w16 h2f"><div class="t m1 x6 h7 y4f ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑠</div></div><div class="c x3e ye w6 h30"><div class="t m0 xc hc y50 ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x3f ye w17 h31"><div class="t m1 x6 h7 y51 ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑠−1</div></div><div class="c x3f ye w8 h32"><div class="t m1 x6 h7 y52 ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x22 ye w9 h2d"><div class="t m0 xc hc y4d ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x14 ye wa h33"><div class="t m0 xc hc y53 ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x14 ye wa h34"><div class="t m0 xc hc y54 ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x40 ye w9 h2d"><div class="t m0 xc hc y4d ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x41 ye we h35"><div class="t m1 x6 h7 y55 ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="c x18 ye wb h36"><div class="t m0 xc hc y56 ff4 fs2 fc0 sc0 ls0 ws0">P</div></div><div class="c x19 ye w13 h37"><div class="t m1 x6 h7 y57 ff4 fs4 fc0 sc0 ls0 ws0">B</div></div><div class="c x25 ye w4 h38"><div class="t m0 xc hc y58 ff4 fs2 fc0 sc0 ls0 ws0">=</div></div><div class="c x26 ye w10 h39"><div class="t m1 x6 h7 y59 ff5 fs4 fc0 sc0 ls0 ws0">2𝑡−𝑟−𝑠−1</div></div><div class="c x27 ye we h3a"><div class="t m1 x6 h7 y5a ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div><div class="c x42 ye w11 h3a"><div class="t m1 x6 h7 y5a ff4 fs4 fc0 sc0 ls0 ws0">=</div></div><div class="c x28 ye w16 h3a"><div class="t m1 x6 h7 y5a ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑠</div></div><div class="c x10 ye w6 h3b"><div class="t m0 xc hc y5b ff5 fs2 fc0 sc0 ls0 ws0">𝐶</div></div><div class="c x11 ye w17 h3c"><div class="t m1 x6 h7 y5c ff5 fs4 fc0 sc0 ls0 ws0">𝑡−𝑠−1</div></div><div class="c x11 ye w8 h3d"><div class="t m1 x6 h7 y5d ff5 fs4 fc0 sc0 ls0 ws0">𝑛−1</div></div><div class="c x12 ye w9 h38"><div class="t m0 xc hc y58 ff4 fs2 fc0 sc0 ls0 ws0">(</div></div><div class="c x13 ye wa h3e"><div class="t m0 xc hc y5e ff4 fs2 fc0 sc0 ls0 ws0">1</div></div><div class="c x13 ye wa h3f"><div class="t m0 xc hc y5f ff4 fs2 fc0 sc0 ls0 ws0">2</div></div><div class="c x14 ye w9 h38"><div class="t m0 xc hc y58 ff4 fs2 fc0 sc0 ls0 ws0">)</div></div><div class="c x15 ye we h40"><div class="t m1 x6 h7 y60 ff5 fs4 fc0 sc0 ls0 ws0">𝑛</div></div></div><div class="pi" data-data='{"ctm":[1.611830,0.000000,0.000000,1.611830,0.000000,0.000000]}'></div></div>

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